Multiple Regression

Multiple Regression in Trial-Based Economic Evaluation

Each coefficient in a multiple regression is a partial effect, read with the other explanatory variables held fixed, which lets a trial's cost or QALY difference be separated from differences in the patients randomised. In health economics it is used to adjust differences in mean costs and mean QALYs between trial arms for baseline covariates, to model skewed cost data and to estimate incremental net benefit from patient-level data. The adjustment matters because a chance imbalance in baseline health can move an incremental cost-effectiveness ratio a long way. The method is often called multivariable regression, while multivariate regression strictly means something else.

One outcome and several predictors

The regression analysis family covers any model of how an outcome varies with explanatory variables, and multiple regression is the case with two or more of them. With a continuous outcome and an identity link it is multiple linear regression, estimated by ordinary least squares as described on that page.

$$y_i = \beta_0 + \beta_1 x_{1i} + \beta_2 x_{2i} + \dots + \beta_k x_{ki} + \varepsilon_i$$

where $y_i$ is the outcome for patient $i$ (a cost, a QALY total or a net benefit), $x_{1i}$ to $x_{ki}$ are the $k$ explanatory variables, $\beta_0$ is the intercept, $\beta_j$ is the coefficient on $x_j$ and $\varepsilon_i$ is the error term.

With several predictors each $\beta_j$ is a partial coefficient, the expected change in the outcome for a one-unit change in $x_j$ among patients who share the same values of the other variables. Hidalgo and Goodman describe the multivariable model as one that can assess independent relationships while adjusting for potential confounders.

Multivariable, multivariate and why the name matters

Hidalgo and Goodman separate two terms that are often swapped. Multivariate analysis has two or more outcome variables; multivariable analysis has one outcome and several predictors. Of 30 articles in a year of the American Journal of Public Health that reported a "multivariate" method, 25 (83%) were in fact multivariable and 5 (17%) were multivariate. They recommend naming models by the number of predictors (simple or multivariable) and by the type of outcome.

A regression of cost on treatment and age is multivariable. A joint model of cost and QALYs, such as seemingly unrelated regression, has two outcomes and is multivariate, as is a multivariate meta-analysis.

Adjusting trial costs and QALYs for baseline covariates

Randomisation balances covariates on average, not in any single trial. The European Medicines Agency guideline on adjustment for baseline covariates (EMA/CHMP/295050/2013) states that adjusting for a covariate strongly or moderately associated with the outcome generally improves efficiency and avoids conditional bias from chance imbalance. It asks for covariates to be pre-specified and few, says the baseline value of a continuous outcome should usually be included, and keeps out of the primary analysis covariates measured after randomisation or chosen for an imbalance seen after the event.

Manca, Hawkins and Sculpher make the matching case for the QALY endpoint of a trial-based cost-effectiveness analysis. Mean baseline utility is invariably imbalanced between arms, and a patient's baseline health state utility value is likely to be highly correlated with QALYs over follow-up, not least because it usually enters the QALY calculation. In their analysis the widely used methods of calculating the QALY difference gave incorrect results whenever baseline utility was imbalanced, whether or not the difference was statistically significant, while multiple regression with baseline utility as a covariate gave appropriate estimates and a measure of their sampling variability.

In a randomised controlled trial the adjusted treatment coefficient keeps its causal reading, while the coefficients on the covariates are only associations conditional on the model. In an observational study the regression deals with confounding only for confounders that were measured and correctly specified, which is why propensity score and instrumental variable methods are often run alongside it.

Worked example with eight patients and an imbalance in baseline utility

All numbers in this example are illustrative. Four patients per arm have a baseline utility, a one-year QALY total and a cost in GBP, and by chance the treated arm starts healthier.

1. Compare raw means. Mean QALYs differ by 0.11 and mean costs by GBP 600, so the unadjusted ICER is 600 divided by 0.11, or about GBP 5,455 per QALY gained. Mean baseline utility differs by 0.10.

2. Fit QALYs on treatment and baseline utility. With a treatment indicator and baseline utility as predictors, the least squares coefficient on baseline utility equals the pooled within-arm slope.

$$\hat{\beta}_U = \frac{S_{ue,1} + S_{ue,0}}{S_{uu,1} + S_{uu,0}}, \qquad \hat{\beta}_T = (\bar{e}_1 - \bar{e}_0) - \hat{\beta}_U (\bar{u}_1 - \bar{u}_0)$$

where $S_{uu,a}$ is the sum of squared deviations of baseline utility from its mean in arm $a$, $S_{ue,a}$ is the sum of their cross-products with the QALY deviations, $\bar{e}_a$ and $\bar{u}_a$ are arm means, and arm 1 is treatment. Baseline deviations are -0.15, -0.05, 0.05 and 0.15 in each arm, so each $S_{uu}$ is 0.05. QALY deviations are -0.11, -0.05, 0.03 and 0.13 with treatment and -0.13, -0.03, 0.05 and 0.11 with control, so each $S_{ue}$ is 0.04.

$$\hat{\beta}_U = \frac{0.04 + 0.04}{0.05 + 0.05} = 0.8, \qquad \hat{\beta}_T = 0.11 - 0.8 \times 0.10 = 0.03$$

where $\hat{\beta}_U$ is QALYs per unit of baseline utility and $\hat{\beta}_T$ is the adjusted difference in mean QALYs. The fitted model is QALYs = 0.05 + 0.03 treatment + 0.8 baseline utility. Baseline utility alone would give a slope of 0.85, because without the treatment indicator part of the arm gap is credited to baseline health.

3. Fit cost on the same predictors. Costs are uncorrelated with baseline utility within each arm here, so the adjusted cost difference stays at GBP 600 and the adjusted ICER is 600 divided by 0.03, or GBP 20,000 per QALY gained.

4. Run a net-benefit regression. At an illustrative GBP 25,000 per QALY, each patient's net benefit is 25,000 times QALYs minus cost. On treatment alone the coefficient is 25,000 times 0.11 minus 600, or GBP 2,150; with baseline utility added it is 25,000 times 0.03 minus 600, or GBP 150. A treatment by baseline utility interaction term is estimated at zero in these data.

5. Interpret. About 0.08 of the apparent 0.11 QALY gain reflects the treated arm's better starting health, so the unadjusted ICER is about a quarter of the adjusted one and the adjusted net benefit is only just positive. A real analysis adds standard errors or bootstrapping intervals.

Cost regressions for skewed data

Patient-level costs are often right-skewed. Mihaylova and colleagues note that least squares on raw costs is unbiased in randomised data but sensitive to extreme values, while a regression on log costs needs back transformation to say anything about mean cost. Barber and Thompson illustrate generalised linear models for trial costs, with an identity link for additive covariate effects or a log link for multiplicative ones; in both of their examples the gaussian model fitted poorly, and with prognostic covariates the choice of distribution could materially change the estimates.

$$\ln E(c_i \mid x_i) = \alpha_0 + \alpha_1 T_i + \alpha_2 x_{2i} + \dots + \alpha_k x_{ki}$$

where $E(c_i \mid x_i)$ is the expected cost of patient $i$ given the covariates, $T_i$ is the treatment indicator, $x_{2i}$ to $x_{ki}$ are baseline covariates and the $\alpha$ terms are log-scale coefficients, fitted for example with a gamma distribution. The exponent of $\alpha_1$ is a ratio of mean costs, so an incremental cost in GBP comes from averaging predicted costs with $T_i$ set to one and to zero.

Manning and Mullahy found that no single log-model estimator was best under all the conditions they simulated and that a less suitable one can lose much precision, so the skewness and tail shape of the data guide the choice.

Costs and effects together

The ICER is a ratio and does not sit naturally as the dependent variable of a regression. Hoch, Briggs and Willan proposed net-benefit regression instead, in which each patient's net benefit is regressed on a treatment indicator and covariates.

$$nb_i = \lambda e_i - c_i = \beta_0 + \beta_1 T_i + \beta_2 x_{2i} + \dots + \beta_k x_{ki} + \varepsilon_i$$

where $\lambda$ is the willingness to pay per unit of effect, $e_i$ and $c_i$ are the patient's effect and cost, and $\beta_1$ is the covariate-adjusted incremental net benefit. Its stated advantages include adjustment for imperfect randomisation and identification of important subgroups. Hoch and colleagues later showed that a positive treatment coefficient means the new treatment is cost-effective at $\lambda$, with an ICER below $\lambda$ when it adds QALYs, and that one minus its one-sided p-value, traced across values of $\lambda$, gives a cost-effectiveness acceptability curve.

Willan, Briggs and Hoch then proposed seemingly unrelated regression, with cost and effect equations fitted jointly. The equations need not share covariates, and the joint fit can be more efficient than two unrelated least squares regressions.

Interaction terms, collinearity and overfitting

Three modelling choices can decide whether the model helps or misleads. An interaction effect term, the product of treatment and a covariate, lets the treatment effect vary with that covariate and is the regression form of a subgroup analysis. The EMA guideline keeps such interactions out of the primary model, so they are best pre-specified and reported as exploration of treatment effect heterogeneity.

When predictors such as age and comorbidity count overlap, multicollinearity inflates the standard errors of their separate coefficients, as collinearity diagnostics show; randomised treatment is unrelated to the covariates in expectation, so its own coefficient is little affected.

Many covariates in a small sample risk overfitting. In simulations by Austin and Steyerberg, linear regression needed about two subjects per variable for coefficients with relative bias below 10% and accurate standard errors, but many more for an unbiased R-squared.

Multiple regression compared with nearby methods

Neighbouring methods share the regression form but differ in outcome and use. The table compares them.

Limitations and common misreadings

Most errors come from reading more into the coefficients than the model can deliver. Five recur.

1. Calling the model multivariate. With one outcome the correct term is multivariable or multiple regression.
2. Choosing covariates after seeing the data. Covariates picked for an observed imbalance belong in the analysis plan instead.
3. Leaving out baseline utility. An unadjusted QALY difference can misstate the ICER even when the imbalance is not significant.
4. Reading every coefficient causally. Only the randomised treatment coefficient has a causal reading in a trial.
5. Assuming adjustment removes all confounding. Outside randomised data, unmeasured confounders remain.

Sources

• Hidalgo B, Goodman M. Multivariate or multivariable regression? American Journal of Public Health. 2013;103(1):39-40. https://doi.org/10.2105/AJPH.2012.300897
• European Medicines Agency, Committee for Medicinal Products for Human Use. Guideline on adjustment for baseline covariates in clinical trials. EMA/CHMP/295050/2013. 26 February 2015. https://www.ema.europa.eu/en/documents/scientific-guideline/guideline-adjustment-baseline-covariates-clinical-trials_en.pdf
• Manca A, Hawkins N, Sculpher MJ. Estimating mean QALYs in trial-based cost-effectiveness analysis: the importance of controlling for baseline utility. Health Economics. 2005;14(5):487-496. https://doi.org/10.1002/hec.944
• Barber J, Thompson S. Multiple regression of cost data: use of generalised linear models. Journal of Health Services Research and Policy. 2004;9(4):197-204. https://doi.org/10.1258/1355819042250249
• Manning WG, Mullahy J. Estimating log models: to transform or not to transform? Journal of Health Economics. 2001;20(4):461-494. https://doi.org/10.1016/s0167-6296(01)00086-8
• Mihaylova B, Briggs A, O'Hagan A, Thompson SG. Review of statistical methods for analysing healthcare resources and costs. Health Economics. 2011;20(8):897-916. https://doi.org/10.1002/hec.1653
• Hoch JS, Briggs AH, Willan AR. Something old, something new, something borrowed, something blue: a framework for the marriage of health econometrics and cost-effectiveness analysis. Health Economics. 2002;11(5):415-430. https://doi.org/10.1002/hec.678
• Hoch JS, Rockx MA, Krahn AD. Using the net benefit regression framework to construct cost-effectiveness acceptability curves. BMC Health Services Research. 2006;6:68. https://doi.org/10.1186/1472-6963-6-68
• Willan AR, Briggs AH, Hoch JS. Regression methods for covariate adjustment and subgroup analysis for non-censored cost-effectiveness data. Health Economics. 2004;13(5):461-475. https://doi.org/10.1002/hec.843
• Austin PC, Steyerberg EW. The number of subjects per variable required in linear regression analyses. Journal of Clinical Epidemiology. 2015;68(6):627-636. https://doi.org/10.1016/j.jclinepi.2014.12.014